Abstract: We develop an algebraic formalism for topological $\mathbb{T}$-duality. More
precisely, we show that topological $\mathbb{T}$-duality actually induces an
isomorphism between noncommutative motives that in turn implements the
well-known isomorphism between twisted K-theories (up to a shift). In order to
establish this result we model topological K-theory by algebraic K-theory. We
also construct an $E_\infty$-operad starting from any strongly self-absorbing
$C^*$-algebra $\mathcal{D}$. Then we show that there is a functorial
topological K-theory symmetric spectrum construction ${\bf K}_\Sigma^{top}(-)$
on the category of separable $C^*$-algebras, such that ${\bf
K}_\Sigma^{top}(\mathcal{D})$ is an algebra over this operad; moreover, ${\bf
K}_\Sigma^{top}(A\hat{\otimes}\mathcal{D})$ is a module over this algebra.
Along the way we obtain a new symmetric spectra valued functorial model for the
(connective) topological K-theory of $C^*$-algebras. We also show that
$\mathcal{O}_\infty$-stable $C^*$-algebras are K-regular providing evidence for
a conjecture of Rosenberg. We conclude with an explicit description of the
algebraic K-theory of $ax+b$-semigroup $C^*$-algebras coming from number theory
and that of $\mathcal{O}_\infty$-stabilized noncommutative tori.

Comments:

18 pages, Dedicated to Professor Marc A. Rieffel on his 75th birthday; v2 added a section on T-duality; v3 added a new section on strongly self-absorbing operads and reorganised the original material; v4 section 1 expanded and a few corrections in section 3; v5 journal reference, DOI, and a bibitem added